Length-preserving monomorphisms for Steenrod algebras at odd primes
In the present paper the authors consider endomorphisms of mod \(p\) Steenrod algebras (the ordinary one \(\mathcal A(p)\), and \(\mathcal B(p)\), the algebra of cohomology operations for the cohomology of cocommutative Hopf algebras over \(\mathbb F_p\)), where \(p\) is an odd prime. It is proved that every such monomorphism which preserves the length of the monomials must be an automorphism (a result which does not hold in the case \(p=2\)). The groups of these length-preserving automorphisms are also determined: \[ \underline{\mathrm{Aut}}(\mathcal A(p))\cong\mathbb Z_{p-1}\times\mathbb Z_{p-1}, \quad \underline{\mathrm{Aut}}(\mathcal B(p))\cong\mathbb Z_{p-1}\times\mathbb Z_{p-1}\times\mathbb Z_{p-1} \] (\(\mathbb Z_{p-1}\) is, as usual, the cyclic group of order \(p-1\)).
scientific article; zbMATH DE number 6991208
scientific article